Definitely the Jayhawks, but only Rainy Day Music. One of my favorite albums--loud guitars and beautiful harmony--but their other stuff seems lackluster.
Ryan Adams, of course. He's tried a lot of different styles, so pick carefully. Some albums: Heartbreaker is excellent acoustic folk/country, Gold is some good pop (though it's a bit too long), and Jacksonville City Nights is Nashville country.
Can't miss Gram Parsons.
A few more: The Louvin Brothers. Ralph Stanley. Jason Eady & The Wayward Apostles. Bright Eyes, I'm Wide Awake, It's Morning. Josh Ritter, The Historical Conquests of ...
These are mostly country/alt-country/folk, the notable exceptions being Ralph Stanley (bluegrass) and Josh Ritter (country-flavoured indie-pop). Other genres are a different question. Not that genre means too much.
I also leave out the obvious older ones: the Carter Family, Johnny Cash (the American recordings!), Patsy Cline, Willie, etc, etc.
Murry told me truly 'bout a girl who's a-moving on.